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# Korovkin type approximation theorem for functions of two variables through statistical A -summability

Author Affiliations

1 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

2 Department of Mathematics, King Abdulaziz University, Jeddah 21589, Kingdom of Saudi Arabia

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Advances in Difference Equations 2012, 2012:65  doi:10.1186/1687-1847-2012-65

 Received: 13 March 2012 Accepted: 23 May 2012 Published: 23 May 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this article, we prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical A-summability. We also study the rate of statistical A-summability of positive linear operators. Finally we construct an example by Bleimann et al. operators to show that our result is stronger than those of previously proved by other authors.

AMS Subject Classification 2000: 41A10; 41A25; 41A36; 40A30; 40G15.

##### Keywords:
density; statistical convergence; A-statistical convergence; statistical A-summability; positive linear operator; Korovkin type approximation theorem

### 1 Introduction and preliminaries

The concept of statistical convergence for sequences of real numbers was introduced by Fast [1] and further studied many others.

Let K ⊆ ℕ and Kn = {k n : k K}. Then the natural density of K is defined by δ(K) = limn n-1 |Kn| if the limit exists, where |Kn| denotes the cardinality of Kn.

A sequence x = (xk) of real numbers is said to be statistically convergent to L provided that for every ε > 0 the set Kε:={k∈ℕ:|xk-L|≥ε} has natural density zero, i.e. for each ε > 0,

In this case we write st- lim x = L. Note that if x = (xk) is convergent then it is statistically convergent but not conversely. The idea of statistical convergence of double sequences has been intruduced and studied in [2,3].

Let A = (ank), n, k∈ℕ, be an infinite matrix and x = (xk) be a sequence. Then the (transformed) sequence, Ax := (yn), is denoted by

where it is assumed that the series on the right converges for each n∈ℕ. We say that a sequence x is A-summable to the limit if yn as n → ∞.

A matrix transformation is said to be regular if it maps every convergent sequence into a convergent sequence with the same limit. The well-known conditions for two dimensional matrix to be regular are known as Silverman-Toeplitz conditions.

In [4], Edely and Mursaleen have given the notion of statistical A-summability for single sequences and statistical A-summability for double sequences has recently been studied in [5].

Let A = (ank) be a nonnegative regular summability matrix and x = (xk) be a sequence of real or complex sequences. We say that x is statistically A-summable to L if for every ε > 0,

So, if x is statistically A-summable to L then for every ε > 0,

Note that if a sequence is bounded and A-statistically convergent to L, then it is A-summable to L; hence it is statistically A-summable to L but not conversely (see [4]).

Example 1.1. Let A = (C, 1), the Cesàro matrix and the sequence u = (uk) be defined by

(1.1)

Then u is is A-summable to (and hence statistically A-summable to 1/2) but not statistically (and not A-statistically as well) convergent.

Let I := [0, ∞) and C(I) denote the space of all continuous real valued functions on I. Let CB(I):= {f C(I): f is bounded on I}.C(I) and CB(I) are equipped with norm

Let Hω(I) denote the space of all real valued functions f on I such that

where ω is the modulus of continuity, i.e.

It is to be noted that any function f Hω(I) is continuous and bounded on I.

The following Korovkin type theorem (see [6]) was proved by Çakar and Gadjiev [7].

Theorem A. Let (Ln) be a sequence of positive linear operators from Hω(I) into CB(I). Then for all f Hω(I)

if and only if

where

Erkuş and Duman [8] have given the stA-version of the above theorem for functions of two variables.Quite recently, Korovkin type of approximation theorems have been proved in [9,10] by using almost convergence; in [11-15] by using variants of statistical convergence and in [16-19] for functions of two variables by using statistical convergence, A-statistical convergence and statistical A-summability of double sequences. In this article, we use the notion of statistical A-summability to prove a Korovkin type approximation theorem for functions of two variables with the help of test functions

### 2 Main result

Let I = [0, ∞) and K = I × I. We denote by CB(K) the space of all bounded and continuous real valued functions on K equipped with norm

Let Hω*(K) denote the space of all real valued functions f on K such that

where ω* is the modulus of continuity, i.e.

It is to be noted that any function f Hω*(K) is bounded and continuous on K, and a necessary and sufficient condition for f Hω*(K) is that

We prove the following result:

Theorem 2.1. Let A = (ank) be nonnegative regular summability matrix. Let (Tk) be a sequence of positive linear operators from Hω*(K) into CB(K). Then for all f Hω*(K)

(2.0)

if and only if

(2.1)

(2.2)

(2.3)

(2.4)

Proof. Since each of the functions belongs to Hω*(K), conditions (2.1)-(2.4) follow immediately from (2.0).

Let f Hω*(K) and (x, y) ∈ K be fixed. Then for ε > 0 there exist δ1, δ2 > 0 such that |f(s, t)-f(x, y)| < ε holds for all (s, t) ∈ K satisfying and .

Let

Hence

(2.5)

Where χD denotes the characteristic function of the set D and . Further we get

(2.6)

Combining (2.5) and (2.6), we get

(2.7)

After using the properties of f, a simple calculation gives that

(2.8)

where

Now replacing Tk(f; x, y) by and taking sup(x,y)∈K, we get

(2.9)

For a given r > 0 choose ε > 0 such that ε < r. Define the following sets

Then from (2.9), we see that D D1 D2 D3 D4 and therefore δ(D) ≤ δ(D1) + δ(D2) + δ(D3) + δ(D4). Hence conditions (2.1)-(2.4) imply the condition (2.0).

This completes the proof of the theorem.

If we replace the matrix A in Theorem 2.1 by identity matrix, then we immediately get the following result which is due to Erkuş and Duman [8]:

Corollay 2.2. Let A = (ank) be nonnegative regular summability matrix. Let (Tk) be a sequence of positive linear operators from Hω*(K) into CB(K). Then for all f Hω*(K)

(2.10)

if and only if

(2.11)

(2.12)

(2.13)

(2.14)

### 3 Statistical rate of convergence

In this section, using the concept of statistical A-summability we study the rate of convergence of positive linear operators with the help of the modulus of continuity. Let us recall, for f Hω*(K)

where

We have the following result:

Theorem 3.1. Let A = (ank) be nonnegative regular summability matrix. Let (Tk) be a sequence of positive linear operators from Hω*(K) into CB(K). Assume that

(i) ,

(ii) st- limn→0 ω*(f; δn) = 0, where

Then for all f Hω*(K)

Proof. Let f Hω*(K) be fixed and (x, y) ∈ K be fixed. Using linearity and positivity of the operators Tk for all n∈ℕ, we have

Hence

Now if we choose then

Therefore

(3.1)

where . Now, for a given r > 0, choose ε > 0 such that ε > r. Let us write

Then E E1 E2 E3 and therefore δ(E) ≤ δ(E1) + δ(E2) + δ(E3). Using conditions (i) and (ii) we conclude

This completes the proof of the theorem.

### 4 Example and the concluding remark

We show that the following double sequence of positive linear operators satisfies the conditions of Theorem 2.1 but does not satisfy the conditions of Corollary 2.2 and Theorem 2.1 of [8].

Example 4.1. Consider the following Bleimann et al. [20] (of two variables) operators:

(4.1)

where f Hω(K), K = [0, ∞) × [0, ∞) and n∈ℕ.

Since

it is easy to see that

Also by simple calculation, we obtain

and

Finally, we get

Now, take A = C(1, 1) and define u = (un) by (1.1). Let the operator Ln : Hω(K) → CB(K) be defined by

It is easy to see that the sequence (Ln) satisfies the conditions (2.1)-(2.4). Hence by Theorem 2.1, we have

On the other hand, the sequence (Ln) does not satisfy the conditions of Theorem A and Corollary 2.2 and Theorem 2.1 of [8], since (Ln) is neither convergent nor statistically (nor A-statistically) convergent. That is, Theorem A, Corollary 2.2 and Theorem 2.1 of [8] do not work for our operators Ln. Hence our Theorem 2.1 is stronger than Corollary 2.2 and Theorem 2.1 of [8].

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

The main result of the paper was proposed and proved by AA, Section 3 was given by MM, and Section 4 was designed by both the authors jointly. All authors read and approved the final manuscript.

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